
Class _i_^34£. 



Book 



Copyright ]^°. 



COPYRIGHT DEPOStC 



I 



ELEMENTARY COURSE IN 
PERSPECTIVE 



ERRATA 

Change the following scales: 

Mg. 10, Scale 1' = 6' to 1" = 6' 
Fig. 11, Scale 1' = 24' to 1'' = 24' 
Fig. 12, Scale 1' = 30' to 1*' = 30' 
Fig. 13, Scale 1' = 8' to 1" = 8' 



ELEMENTARY COUESE IN 
PEESPECTIYE 



By 

SHERMAN M. TURRILL 

Civil Engineer, Assoc. Am. Soc. C. E, 



WITH 16 ILLUSTRATIONS AND FOLDING PLATES 




NEW YORK 
D. VAN NOSTRAND COMPANY 
23 Murray and 1910 27 Warren Sts. 






Copyright, 1910 
By D. Van Nostrand Co. 



The Plimpton Press Norwood Mass. U.S.A. 



CCI.A253!J34 



PREFACE 

In the following pages the author has en- 
deavored to illustrate the mechanical appUcation 
of the principles of Descriptive Geometry to the 
making of a perspective drawing. Two methods 
are used: the ^'Method by the Use of the Plan/' 
where the orthographic projections of rays are 
used, and the ^'Method by Scale." In the former, 
all details should be drawn with instruments, 
especially if there is much mechanical labor en- 
tailed. On the other hand, the ^^ Method by 
Scale'' should serve the purpose, and is, in fact, 
the quicker process, where the accurate location 
of the leading or important points only is desired. 
The first method is discussed mainly for the 
assistance of the mechanical draftsman, whereas 
the second is intended for the artist and free- 
hand draftsman. 

In writing this treatise the author has assumed 
that the student is more or less familiar with 
Descriptive Geometry. The subjects taken have 



vi PREFACE 

been selected with a view toward illustrating 
important principles and for their general in- 
terest. 

In addition will be found a large number of 
problems covering these principles as applied to 
the illustrated subject, in a variety of positions. 

Shekman M. Turrill. 

Chicago, 111., 
January, 1910. 



\ 



TABLE OF CONTENTS 
PART I 

METHOD BY ORTHOGRAPHIC PROJECTION 

CHAPTER I 

General Description of Perspective 

ART. PAGE 

1. Definitions and Descriptions 3 

2. Notation 5 

CHAPTER II 

Parallel Perspective 

3. Pictorial Perspective of a Pyramid 7 

4. Perspective of a Pyramid 9 

5. Perspective of a Building 10 

6. Problems 12 

CHAPTER III 

Angular Perspective 

7. Perspective of a Cube and Parallelopiped ... 17 

8. Problems 21 

CHAPTER IV 
Perspective of Curves 

9. Perspective of a Plane Curve 26 

10. Perspective of a Cylinder 27 

11. Problems ,,,,,, 28 

vii 



viii CONTENTS 

CHAPTER V 

Perspective of Shadows 

ART. PAGE 

12. Perspective of the Shadow of the Frustum of a Pyra- 

mid by use of Horizontal Proj ection of the Shadow 29 

13. Perspective of the Shadow of the Frustum of a Pyra- 

mid by the Method of the Vanishing Point of Rays 31 

14. Perspective of the Shadow of a Cube and Parallelo- 

piped 33 

15. Perspective of the Shadow of the Frustum of a 

Pyramid on an Obhque Plane 37 

16. Problems 45 



PART II 
METHOD BY SCALE 

CHAPTER VI 

General Description of the Method 

17. Comparison of the Two Methods 49 

CHAPTER VII 
Parallel Perspective 

18. Perspective of a Pyramid . 52 

19. Perspective of a Building 57 

20. Problems 62 

CHAPTER VIII 

Angular Perspective 

21. Perspective of a Cube and Parallelopiped ... 63 

22. Problems 66 



CONTENTS ix 

CHAPTER IX 

Perspective of Curves 
ART. page 

23. Perspective of a Cylinder 67 

24. Problems 69 

CHAPTER X 

Perspective of Shadows 

25. Perspective of the Shadow of a Cube and Parallele- 

piped 70 

26. Problems 71 

ILLUSTRATED FIGURES 
PART I 
Method by Orthographic Projection facing 

FIG. PAGE 

1. Pictorial Perspective of a Pyramid 10 

2. Parallel Perspective of a Pyramid 10 

3. Parallel Perspective of a Building 16 

4. Angular Perspective of a Cube and Parallelopiped . 26 

5. Perspective of a Circle and Cylinder 28 

6. Perspective of the Shadow of a Frustum of a Pyra- 

mid 32 

7. Perspective of the Shadow of a Cube and Parallelo- 

piped 46 

8. Perspective of the Shadow of the Frustum of a 

Pyramid on an Oblique Plane 46 

PART II 
Method by Scale 

9. Orthographic Projection of a Pyramid . . . page 53 
10. Parallel Perspective of a Pyramid . . facing page 5§ 



X CONTENTS 

FACING 
FIG. PAGE 

11. Orthographic Projection of a Building .... 62 

12. Parallel Perspective of a Building 62 

13. Orthographic Projection of a Cube and Parallele- 

piped 66 

14. Angular Perspective of a Cube and Parallelopiped 66 

15. Orthographic Projection of a Cylinder . . . page 67 

16. Perspective of a Cylinder facing page 70 



PART I 

PERSPECTIVE BY ORTHOGRAPHIC 
PROJECTION 



Chapter I 
PERSPECTIVE 

I. The perspective of any object is its projec- 
tion on a vertical plane when the point of sight 
is at a finite distance from the object and the 
plane of projection, which is commonly called 
the picture plane, as ahcd, Fig. 1. 

Point of sight or station is the point from which 
the object is viewed and is the apex of a cone 
from which all rays from different points of the 
object have their origin, as S, Fig. 1. Hence the 
name conical projection, which is sometimes given 
as the definition of perspective. 

Picture plane is the vertical plane upon which 
the perspective is drawn and is usually between 
the object and the point of sight, as P. P., Fig. 1. 
If it is desired to have the perspective larger than 
the object the picture plane is placed behind the 
object and the rays of light pass from the point 
of sight through the object to the picture plane 
behind. 

The horizon is the line of intersection of the 
picture plane with a horizontal plane through 

3 



4 ELEMENTARY COURSE IN PERSPECTIVE 

the point of sight, as shown in Fig. 1. These 
two planes are always perpendicular to each 
other. 

The center of picture is the orthographic pro- 
jection of the station on the picture plane, as s", 
Fig. 1. 

The vanishing point of any line or system of 
Hnes is found by passing a line through the station 
parallel to the line or system of hnes, and the 
point where this hne intersects or pierces the 
picture plane is the vanishing point of the line 
or system of hnes. In other words it is the 
perspective of infinity, the point where the parallel 
hnes of any system meet. If the Hnes are hori- 
zontal their vanishing point is in the horizon, as 
V. P., Fig. 1. If not horizontal the vanishing 
point is above or below the horizon as (V. P.)^ in 
P. P., Fig. 1. 

By indefinite perspective of a hne is meant 
the perspective of a line which extends from the 
picture plane to infinity, c" V. P., Fig. 1, is the 
perspective of such a hne, c" being the perspective 
of the point of the hne CB in the picture plane, 
and V. P. being the perspective of that point of 
the hne CB which is at infinity. Fig. 1. 

The definite perspective of a line is the per- 
spective of a definite portion of an indefinite 
perspective and is any portion of the indefinite 



PERSPECTIVE 5 

perspective. Thus the perspective he is a portion 
of the indefinite perspective c" V. P., and hence 
the definite perspective of B C, Fig. 1. 

Parallel perspective of any object is the per- 
spective of that object when at least one of the 
faces of a parallelopiped enclosing the object is 
in or parallel to the picture plane as shown in 
Fig. 1. pictorially and in Figs. 2 and 3 ortho- 
graphically. 

NOTATION 

II. Station or Point of Sight is represented 

by S. 

Picture Plane is represented by P.P. 

Horizon is represented by Hor. 

Horizontal Plane is represented by H. 

Vanishing Point is represented by V. P. 

Points themselves are represented by capital 
letterS; and their projections by small letters with 
the power h. or v. designating horizontal or verti- 
cal projections respectively. 

Lines are represented by two points. 

Planes are named by a capital letter, and rep- 
resented by their traces, these traces being let- 
tered with the prefix H. or V. placed before the 
capital letter of the plane to designate horizontal 
and vertical trace of the plane. 



6 ELEMENTARY COURSE IN PERSPECTIVE 

Where a line lies in an oblique plane, it will 
intersect the horizontal and vertical planes of 
projection if produced, and where it intersects 
these planes of projection is lettered h. i. and v. i. 
respectively. 

Center of picture is the vertical projection of 
the station. 



Chapter II 
PARALLEL PERSPECTIVE 

III. In the pictorial illustration, Fig. 1, let 
ABODE represent a pyramid in space, one of the 
faces c V C D enclosing it being parallel to the pic- 
ture plane. The orthographic projection of this 
object in the horizontal plane H. is aVc'' etc. 
Its vertical projection in the picture plane P. P. 
is aVd". The horizontal projection of S, the sta- 
tion, is s^, and of the rays S Bj S D, etc., is sV, 
sV, etc., which intersect XX in 6'',^'', etc. From 
these points erect perpendiculars to H. and they 
he in the picture plane P. P. These last hnes, if 
produced far enough, will intersect their corre- 
sponding rays in h, d, etc., which are the perspec- 
tive of the points 5, Z), etc. If these points in 
the perspective are joined by lines in the same 
sequence as the original object we will have the 
perspective abed of the object ABODE, 

The method by vanishing points is the one used 

in practice and will now be described by the use 

of Fig. 1. The steps are the same as above up to 

and including the perpendiculars in P. P. In 

7 



8 ELEMENTARY COURSE IN PERSPECTIVE 

order to obtain the vanishing point (V. P.) pass 
through the station S, a straight hne parallel to 
CB and DE. The horizontal projection of this 
hne through S is sV parallel to cV.^ From s* 
erect a perpendicular to H. intersecting the 
parallel hne from S in s". This is where the 
parallel line through S intersects the picture plane 
P. P., and is the vanishing point of the system 
of lines parallel to CB, and is the perspective of 
infinity where these parallel hnes meet. The line 
C B, if produced, intersects the picture plane in 
c", and the indefinite perspective of the hne C B 
is c" V. P., and the definite perspective of that 
portion of the line from C to 5 is cb, found by 
dropping perpendiculars in P. P. from h'' and c* 
until they intersect the indefinite perspective 
c'' V. P. in 6 and c. The same method of reason- 
ing is followed for the line DE and a line through 
A parallel to D E, and thus a and d are found. 
To complete the perspective, lines are drawn 
from do d, d to a, c to a, and h to a, and we have 
the perspective abed of the object A B C D. As 
a check on the work cd should be parallel to 
c^d"" and XX since C D is parallel to H. and P. P., 
and therefore a hne through S parallel to C D 
meets P. P. at infinity, and therefore the per- 

* If two lines are parallel in space their corresponding projec- 
tions are parallel. 



PARALLEL PERSPECTIVE 9 

spective of C i) is parallel to itself. As another 
check we can find the vanishing point of A C. 
It is found by passing a fine through S parallel to 
A C, and where it intersects P. P. is the vanishing 
point of A C. To find this pass through the 
horizontal and vertical projections of S, i.e., s^ 
and s"", straight fines sV, s" (V. P.)^ parallel to 
the corresponding projections of A C, i.e., aV, 
aV. 

From o"" in X X erect a perpendicular to H. in 
P. P. and it intersects the fine through s"" in 
(V. P.)^, which is the vanishing point of A C. 
Hence when c is determined, draw the fine c 
(V. P.)^ and it should coincide with ca, and fike- 
wise we can find the vanishing points oi A D 
and A E. 

IV. The orthographic projection of the per- 
spective abed is shown in Fig. 2. The same 
points bear the same letters as in Fig. 1, so 
that the above description for the method of 
vanishing points will describe this figure, only 
bearing in mind that the horizontal and vertical 
planes of Fig. 1 are so swung that the part of H. 
in front of X X and that part of P. P. below 
XX coincide and are below X X while that part 
of H. back of X X and that part of P. P. above 
X X also coincide, and are above X X as shown 
in Fig. 2. 



10 ELEMENTARY COURSE IN PERSPECTIVE 

V. Figure 3 gives the orthographic projection 
of a more comphcated problem of parallel per- 
spective than was found in Fig. 2. The X X 
axis or horizontal projection of the picture plane 
and the plan of the building aVfrnl" are located 
so that one end of the building is parallel to the 
picture plane. The horizontal projection of the 
station point s^ is so chosen at random that a 
good perspective is obtained, and the horizontal 
projection of the rays sV, sV, s^d^, etc., are drawn 
intersecting the X X axis in a^, c^, (fj etc., respec- 
tively. The edges of the building or lines AM,PLj 
C J, etc., are produced until they intersect the 
picture plane and the vertical intersection of these 
lines are shown in a", p"j c'', etc., respectively. 
The points are connected as shown, and we get 
the end of the building in its true shape and size. 
The vanishing point, of the system of parallel 
lines AM, PL, C J, etc., is found by passing a 
Hne through S parallel to them and where it 
intersects the picture plane is the vanishing point. 
We therefore draw a hne through s'', parallel to 
a^m^j pV, etc., intersecting the XX axis in s"", 
and the perpendicular from s* to the intersection 
with the horizon gives V. P., since the system of 
lines is horizontal. The indefinite perspective of 
the above system is then found or shown by 
a" V. P., p" V. P.; c" V. P., etc. The definite 





// :l / 



6 ' c,,y d J:o' PICTURE IfLANE i 



1^-P/. 



^ 



PICTURE PLANE 




PARALLEL PERSPECTIVE 11 

perspectives of these lines are then found by 
dropping perpendiculars from a"", {nf not shown), 
p^j Vy d", f, etc., until they intersect their re- 
spective indefinite perspectives, and that portion 
included between the respective points is the 
definite perspective a (m not shown), plj c j, etc. 
To get the perspective of R draw the horizontal 
projection of a ray from r^, and where this ray 
intersects the X X axis in r^ erect a perpendicular, 
and where it intersects the indefinite perspective 
f V. P. will be the perspective of R represented 
by r. Join cr and we have the perspective cr of 
C R. The perspectives of A, J K, etc., are 
found in the same way. The perspectives of 
PR, P Oj etc., are found by first obtaining the 
perspective p and then join r and p, also o and p, 
etc. The perspectives or, cc^, aa^, etc., are all 
perpendicular or parallel to the horizon, for hori- 
zontal or perpendicular lines parallel to the pic- 
ture plane bear the same relation to the horizon 
in perspective. 

The perspective of the wing is found as follows. 
Produce a horizontal line through F and / until 
it intersects the picture plane and is represented 
by /"". The indefinite perspective of FI is /"" 
(V. P.), and the definite perspective fi is found 
as described for other lines. The perspective fd 
is a line parallel to the horizon, and the perspective 



12 ELEMENTARY COURSE IN PERSPECTIVE 

d is found by drawing a ray dh^y and where it 
intersects X X in d'' drop a perpendicular, and 
where it crosses the horizontal hne through / 
gives d. By following these principles for each 
point the entire perspective is constructed. The 
perspectives op, pr, re, etc., could be obtained by 
finding the vanishing points of these lines, and 
as an illustration of this method we will take the 
line R P. r^p^; fp" are the horizontal and ver- 
tical projections of R P. Therefore to find the 
vanishing point of this Hne pass through the 
station a hne parallel to R P. The projections 
of this line through the projections of the station 
are sV, sT, and will be parallel to the projections 
r^p^; r'"'p", respectively. Where this hne S T inter- 
sects the picture plane will be the vanishing point 
of the line R P, and it is represented by (V. P.)^, 
and the perspective oi RP lies on the line r 
(V. P.)^ and coincides with the hne rp already 
drawn. In like manner the perspective of any 
line in the drawing can be found. 

PROBLEMS IN PARALLEL PERSPECTIVE 

VI. 1. Find the parallel perspective of Fig. 3 
when the end of the building is in the picture 
plane. 

2. Find the perspective of Fig. 3 when the 
building is turned through 90° and FI is parallel 



PARALLEL PERSPECTIVE 13 

to the picture plane, and find the vanishing point 
of at least one system of obhque fines. 

3. Find the parallel perspective of Fig. 3 when 
FI is in the picture plane, and find the vanishing 
point of at least one system of oblique fines. 

4. Find the perspective of Fig. 3 when the 
station is moved as far to the left as it is to the 
right of the center of the picture. 

5. Find the parafiel perspective of Fig. 3 when 
the wing is removed and the end of the roof has 
no triangular sloping surface, and find the vanish- 
ing point of at least one system of oblique lines. 

6. Find the parallel perspective of Fig. 4 when 
A B is parallel to the picture plane. 

7. Find the parallel perspective of Fig. 4 when 
A B is in the picture plane, and find the vanish- 
ing point of at least one system of oblique lines. 

8. Find the parallel perspective of Fig. 4 when 
BC is parallel to the picture plane, and find the 
vanishing point of at least one system of obfique 
fines. 

9. Find the perspective of Fig. 4 when B C is 
in the picture plane. 

10. Find the perspective of Fig. 4 when A B 
is parallel to the picture plane and the parallelo- 
piped EI rests on the right side of the cube or on 
the edge B C. 

11. Find the perspective of Fig. 4 when A B 



14 ELEMENTARY COURSE IN PERSPECTIVE 

is parallel to the picture plane and the block EI 
rests on the edge A B, and the entire object is 
behind the picture plane; and find the vanishing 
point of at least one system of obhque fines. 

12. Find the perspective of Fig. 4 when an 
edge of the cube is parallel to the picture plane, 
but the object is in front of the picture plane 
instead of behind it. 

13. Find the perspective of Fig. 3 when the 
object is in front of the picture plane, and find 
the vanishing point of at least one system of 
obfique fines. 

14. Find the perspective of a right parallelo- 
piped when one face is parallel to the picture 
plane and one end is the base of the object. 

15. Find the perspective of a right parallelo- 
piped when one face is parallel to the picture 
plane and one face as the base of the object, the 
ends being both perpendicular to the base and 
the picture plane. 

16. Find the perspective of a right parallelo- 
piped when the end is parallel to the picture plane. 

17. Find the perspective of a right parallelo- 
piped when one face is in the picture plane. 

18. Find the perspective of a right octagonal 
prism when one edge is parallel to the picture 
plane and the octagon is the base. 

19. Find the perspective of a right octagonal 



PARALLEL PERSPECTIVE 15 

prism when one face is parallel to the picture 
plane and the octagon is the base. 

20. Find the perspective of a right octagonal 
prism when one edge is in the picture plane and 
the octagon is the base. 

21. Find the perspective of a right octagonal 
prism when one edge is parallel to the picture 
plane and one of the rectangular faces is the base, 
and find the vanishing point of at least one system 
of obUque Unes. 

22. Find the perspective of a right octagonal 
prism when one face is parallel to the picture 
plane and an edge is the base. 

23. Find the perspective of a right octagonal 
prism when one face is in the picture plane and 
an edge is the base, and find the vanishing point 
of at least one system of obUque lines. 

24. Find the perspective of a right octagonal 
prism when one edge is in the picture plane and 
one of the rectangular faces is the base. 

25. Find the perspective of a parallelopiped 
block with a mortise in one face and one end of 
the sohd is halved. 

26. Find the parallel perspective of a parallelo- 
piped block when one end has a tenon and the 
other square. 

27. Find the perspective of a right square 
pyramid. 



16 ELEMENTARY COURSE IN PERSPECTIVE 

28. Find the perspective of an oblique square 
pyramid and find the vanishing point of at least 
one system of obhque Hnes. 

29. Find the perspective of a tee made up of 
two parallelopipeds of the same size and with 
square cross-sections. 

30. Find the perspective of the frustum of an 
oblique square pyramid^ and find the vanishing 
point of at least one system of oblique hnes. 



PICTURE PLANE 



r 



^kl 



Is' 

(V.P.) HORIZON 



^'^'^ 



^Sh 



hG. 3. 



Chapter III 
ANGULAR PERSPECTIVE 

VII. Figure 4 represents the angular perspec- 
tive of a cube with a parallelepiped block resting 
against it. The object is placed entirely back 
of the picture plane and at any convenient dis- 
tance from it, and according to the definition of 
angular perspective the vertical faces are placed at 
oblique angles with the picture plane as shown in 
the plan (^h^&^. The vertical projection is shown 
by a^'lfc'^e^' and the end projection by a'h'fg\ 

The station S is selected at random so as to get 
as good a perspective as possible, its horizontal 
projection being denoted by s^ and its vertical by 
s"" on the horizon, which latter is the intersec- 
tion of the picture plane with a horizontal plane 
through the station or point of sight. The van- 
ishing points of the four systems of lines, 1st, B C, 
GK, etc.; 2d, AB, etc.; 3d, EG, etc.; 4th, E F, 
IK, etc.; are found by passing through the sta- 
tion Unes parallel to these systems, as sVj, par- 
allel to 6V and s" (V. P.)^ parallel to &V; sh^ 
parallel to a^h^ and s^ (V. P.)^ parallel to a^6"; sVj 

17 



18 ELEMENTARY COURSE IN PERSPECTIVE 

parallel to eY and s" (V. P.)' parallel to eY; 
sVa parallel to iV and sV parallel to iV,"^ re- 
spectively; and where they intersect the picture 
plane are the vanishing points of the respective 
systems, as (V. P.)^ ; (V. P.)' ; (V. P.)^ etc. The 
first two systems are horizontal lines and hence 
the vanishing points must be on the horizon. 
The last two systems are obhque Hnes, and the 
vanishing point must be above or below the 
horizon according as to whether the system of 
lines recede from the picture plane in ascending 
or descending. In this perspective the horizontal 
projections of the rays sV, etc., are drawn as in 
parallel perspective, intersecting the XX axis 
or horizontal projection of the picture plane in 
a*, h"", 6*, etc. From these points perpendiculars 
to X X are drawn and on these lines somewhere 
are the perspectives of the points A, B, E, etc. 
In order to locate these points we must first find 
the indefinite perspective of fines containing these 
points and the most convenient lines will be hori- 
zontal fines. Therefore pass a horizontal fine 
through B, being one of the lines of the system 
B C, G K, etc., and hence coinciding with B C. 
The indefinite fine B C extends from the picture 
plane to infinity. Its intersection with P. P. 

* If straight lines are parallel in space their corresponding pro- 
jections are parallel. 



ANGULAR PERSPECTIVE 19 

being found by drawing a horizontal line through 
h^c^ until it intersects the X X axis at rf, then 
erect a perpendicular to XX, also draw a hori- 
zontal Une through h% and where it intersects the 
perpendicular is the point where B C intersects 
the picture plane denoted by n". This perpen- 
dicular nV is called a measuring Une because all 
lines in the face of the cube B C intersect the 
P. P. in this hne rfn". Therefore the indefinite 
perspective of all horizontal lines lying in face 
B C are drawn from points in nV or nV produced 
to (V. P.)S and the indefinite perspective of the 
line 5 C is represented by n" (V. P.)S and the defi- 
nite perspective of the line 5 C is found as in 
parallel perspective and shown by he. The per- 
spective of the bottom edge of the cube parallel 
to 5 C is found exactly hke h c. The hne A B 
being horizontal and of the second system we 
know that its vanishing point is (V. P.)^ and the 
perspective of B is h, therefore the perspective 
oi A B must be on the line h (V. P.)^ and the 
perspective of a is located as at c. The perspec- 
tive of the cube is completed by the repetition of 
the methods described in this article or those 
under parallel perspective. The perspective of 
the parallelopiped is now to be described. By 
passing horizontal lines through E, F, and G, 
they intersect the picture plane in the points 



20 ELEMENTARY COURSE IN PERSPECTIVE 

r\, rl, r", respectively, and the line rh^^ is a measur- 
ing line for all heights in the plane F G. The 
indefinite perspective of these horizontal Unes are 
rl (V. P.)', n (V. P.)', r (V. P.)', since they are of 
the system A B, etc. The perspectives e and / 
are found in the same manner as a and c. The 
perspective of E G, etc., is of the third system, 
and its vanishing point is (V. P.)^ Its indefi- 
nite perspective is found by finding where E G 
produced intersects the picture plane as v i 
and connecting it with (V. P.)^ On this indefi- 
nite perspective v i (V. P.)^ are the perspectives 
of E and G. E having already been found at e 
where the perpendicular from e'' intersects the 
indefinite perspective rl (V. P.)^ which in turn 
lies on the indefinite perspective v i (V. P.)^ 
The perspective g of the point G is found from 
gV and a perpendicular to X X from ^*, and 
where it intersects the indefinite perspectives 
r" (V. P.)^ and v i (V. P.)^ The perspectives of 
the other fines of the third system are found in a 
similar manner, and the points e and /; i and kj 
etc., are connected and the perspective ei is com- 
pleted. As a check the fines ef, ki, etc., of the 
fourth system should meet at (V. P.)^ Another 
check is to find the perspectives of g, k, i, etc., as 
e and /were found, and see if the fines joining the 
proper points vanish to (V. P.)^ or (V. P.)^ By 



ANGULAR PERSPECTIVE 21 

following the above method the entire perspective 
can be completed. 

PROBLEMS IN ANGULAR PERSPECTIVE 

VIII. 1. Find the perspective of Fig. 3 when 
one edge of the building is behind the picture 
plane, and find the vanishing point of at least one 
system of obhque hnes. 

2. Find the perspective of Fig. 3 when one 
edge is in the picture plane. 

3. Find the perspective of Fig. 3 when the 
wing is removed and the end of the roof has no 
triangular sloping surface. One edge is behind 
the picture plane and the point of sight to the 
right of the object, find the vanishing point of at 
least one system of obhque lines. 

4. Find the perspective of Fig. 3 when the 
wing is removed and the end of the roof has no 
triangular sloping surface. One edge behind the 
picture plane and the point of sight to the left of 
the object, and find the vanishing point of at 
least one system of obhque hnes. 

5. Find the perspective of Fig. 4 when the 
vertical edge through B is in the picture plane. 

6. Find the perspective of Fig. 4 when block 
EI rests on edge A B and E is in the picture plane, 
and find the vanishing point of at least one sys- 
tem of oblique lines. 



22 ELEMENTARY COURSE IN PERSPECTIVE 

7. Find the perspective of Fig. 4 when block 
EI rests on edge A B and E is behind the picture 
plane. 

8. Find the perspective of Fig. 4 when block 
EI rests on edge C D and vertical edge through 
B is behind the picture plan^, and find the vanish- 
ing point of at least one system of obUque fines. 

9. Find the perspective of Fig. 4 when block 
EI rests on edge C D and vertical edge through 
B is in the picture plane. 

10. Find the perspective of Fig. 4 when block 
EI rests on edge B C and vertical edge through B 
is parallel to the picture plane. The whole object 
being behind the picture plane, find the vanishing 
point of at least one system of obfique fines. 

11. Find the perspective of Fig. 4 when block 
EI rests on edge B C and its corner nearest to the 
picture plane in problem 10 is in the picture plane 
in this problem, and find the vanishing point of 
at least one system of obfique fines. 

12. Find the perspective of Fig. 4 when the 
entire object is in front of the picture plane, and 
find the vanishing point of at least one system of 
oblique lines. 

13. Find the perspective of a right parallelo- 
piped when the object rests on its end one edge 
being in the picture plane. 

14. Find the perspective of an oblique parallelo- 



ANGULAR PERSPECTIVE 23 

piped when the object rests on its end one edge 
being behind the picture plane, and find the van- 
ishing point of at least one system of obhque Unes. 

15. Find the perspective of a right parallelo- 
piped when the object rests on one of its faces and 
one edge is behind the picture plane and the end 
perpendicular to the base. 

16. Find the perspective of a right octagonal 
prism when one edge is in the picture plane and 
one end is the base, and find the vanishing point 
of at least one system of oblique lines. 

17. Find the perspective of a right octagonal 
prism when one corner is in the picture plane and 
one face is the base. 

18. Find the perspective of a right octagonal 
prism when one edge is behind the picture plane 
and one edge is in the base, and find the vanish- 
ing point of at least one system of obhque fines. 

19. Find the perspective of a right octagonal 
prism when one corner is behind the picture plane 
and one face is the base, and find the vanishing 
point of at least one system of obfique fines. 

20. Find the perspective of a right octagonal 
prism when no faces are parallel to the picture 
plane and the octagon is the base, one of the 
edges being parallel to the picture plane, and 
find the vanishing point of at least one system of 
obfique fines. 



24 ELEMENTARY COURSE IN PERSPECTIVE 

21. Find the perspective of a parallelepiped 
block with a mortise in one face and one end of 
the solid is halved, one edge being parallel to the 
picture plane. 

22. Find the perspective of a parallelopiped 
block when one end has a tenon and the other 
end square, one edge being parallel to the picture 
plane. 

23. Find the perspective of a right square 
pyramid, one corner behind the picture plane. 
Find the vanishing point of at least one system 
of obhque lines. 

24. Find the perspective of a right square 
pyramid when one corner is in the picture plane. 

25. Find the perspective of an obhque square 
pyramid, and the vanishing point of one system 
of obhque Unes. 

26. Find the perspective of an obhque octagonal 
pyramid, and the vanishing point of at least one 
system of obhque lines. 

27. Find the perspective of a tee made of two 
parahelopipeds of the same size and with square 
cross-sections, one edge in the picture plane. 

28. Find the perspective of a tee made of two 
parahelopipeds of the same size and with square 
cross-sections, one edge being behind the picture 
plane. 

29. Find the perspective of the frustum of an 



ANGULAR PERSPECTIVE 25 

oblique square pyramid. Find the vanishing 
point of one system of oblique lines. 

30. Find the perspective of a triangular prism 
and the vanishing point of one system of obUque 
lines. 



Chapter IV 
PERSPECTIVE OF CURVES 

IX. The perspective of a curve is found by the 
methods of parallel or angular perspective. In 
general the former will be the simpler method as 
shown in Fig. 5. In this figure the curve chosen 
to find the perspective of is a circle whose hori- 
zontal projection is shown as a^6V^^, etc., and 
vertical projection as aV. The picture plane, or 
XX axis, horizon, station, and vanishing point 
are located as described in parallel perspective. 
The station having been determined we pass rays 
from it to a finite number of points on the curve, 
their horizontal projection being denoted by sV; 
sV; sV; sy^, etc., always being sure to draw the 
extreme rays S D and S I whose horizontal pro- 
jections, sV and sV, are tangent to the curve. 
These projections of the rays cross the X X axis 
in a"", ¥, e% f", d"", and i"", respectively. Through 
the points A, B, E, etc., of the curve from which 
rays are drawn, pass horizontal hues perpendicu- 
lar to the picture plane, their horizontal projec- 
tions being shown by lines through a^, V", e^, etc., 

26 








(V.PJ 



PERSPECTIVE OF CURVES 27 

perpendicular to the X X axis. Their front pro- 
jections are shown in the points a", If, e"", etc., 
respectively. The vanishing point of these hori- 
zontal Hnes is V. P., which coincides with s". 
The indefinite perspective of these horizontal hnes 
is a^V. P.; 6"V. P.; e^V. P., etc., respectively. 
From the points a*, 6*, e"", etc., on the X X axis 
draw Hnes perpendicular to the same, and where 
they intersect the indefinite perspectives a^ V. P. ; 
h" V. P.; e^V. P., etc., respectively, in a, h, e, etc., 
is the perspective of these points of the curve. 
By this method as many points on the curve as 
are necessary can be obtained. 

X. If we wish to find the perspective of a right 
cyUnder as shown in Fig. 5, find the perspective 
of another curve whose horizontal projection co- 
incides with the first curve and whose vertical 
projection is a^^'^y in the same manner as in the 
first curve, as afiie^, etc. In order to complete 
the perspective of the cyHnder, vertical fines have 
to be drawn from the extreme points of the per- 
spectives of the circles as lit, and dd^. These 
points are the perspectives of the points I, D, I^, 
Di, and in horizontal projection are the points 
i^, d^j where the rays are tangent to the curve. 
By this method the perspective of any curved 
object can be obtained. 



28 ELEMENTARY COURSE IN PERSPECTIVE 

PROBLEMS IN PERSPECTIVE OF CURVES 

XI. 1. Find the perspective of a right circular 
cone. 

2. Find the perspective of a right frustum of 
a circular cone. 

3. Find the perspective of an oblique cylinder 
with an elliptical base. 

4. Find the perspective of an oblique frustum 
of a cone whose right section is an ellipse. 

5. Find the perspective of an obhque cone 
whose right section is a circle. 

6. Find the perspective of the intersection of 
a right circular cylinder with an oblique cone 
having its right section a circle. 

7. Find the perspective of a sphere. 



Chapter V 
PERSPECTIVE OF SHADOWS 

XII. Fig. 6 represents the perspective gajjfjd 
of the shadow of the frustum of a square pyramid. 
The perspective abcg of the object is first drawn by 
the methods aheady outUned in this part. The 
shaded portion g^a^bic^^,e^ is the horizontal projec- 
tion of the shadow of the frustum on the horizon- 
tal plane of its base, and is drawn by the use of 
the horizontal and vertical projections of the rays 
of Ught aided by the plan a^hVg^ and the eleva- 
tion a%\Y- 

The perspective gashsC,e of this shadow g^a]h]c]e^ 
and the method of obtaining it will now be ex- 
plained. Since the orthographic projection of this 
shadow is in the horizontal plane all of its bound- 
ing Hues, g^ aj; a]h], etc., and the horizontal 
projections of the rays a^a]] hj^b^j etc., are all 
horizontal hues, and therefore have their vanish- 
ing points in the horizon. 

Since the horizontal projections of all the rays 
aj^ a^; h^h], etc., are parallel they will have the 
same vanishing point. This is obtained by pass- 

29 



30 ELEMENTARY COURSE IN PERSPECTIVE 

ing through the station a Une parallel to the 
system of Hnes ay^a^] h^h^, etc., and where it inter- 
sects the picture plane is the vanishing point of 
that system. 

We therefore pass through the horizontal and 
vertical projections of the station s^ and s", re- 
spectively/ a Hne parallel to the respective pro- 
jection of the system of Hnes a^a^^', i^i^b^, and 
obtain by descriptive geometry the vanishing point 
(V. P.)* of this system. The indefinite perspective 
of the lines aM] h%i, etc., is a, (V. P.)^; 6l(V.P.)^ 
etc., respectively. The definite perspectives of 
these lines respectively are a^as] hibs, etc., found by 
drawing horizontal projections of rays a^s^; 6js*, 
etc., and where they intersect the XX axis in 
aj; 6 J, etc., drop perpendiculars, and where they 
intersect the indefinite perspective of the corre- 
sponding Hnes we get respectively a^, hs, etc. To 
complete the perspective join g, as] as, 6,; hs, c^ 
and Csy e, and we have the shaded portion as the 
visible portion of the perspective of the shadow 
of the frustum of the pyramid. 

As a check to this perspective, find the vanish- 
ing points (V. P.)^ for Hne aj6?, which is paraUel 
to Hne a^lf", etc., and hence has the same vanish- 
ing point; (V. P.)^ for line gVs] (V. P.)^ for line 
cj6j, which is parallel to Hne d'h^, etc., and hence 
has the same vanishing point; (V. P.)^ for Hne 



PERSPECTIVE OF SHADOWS 31 

e^c^. With these vanishing points found draw 
the indefinite perspective g (V. P.)^ and as a 
check it should pass through a^; next through a^ 
draw the indefinite perspective a^ (V. P.)^ and 
as a check it should pass through 6„ again the 
indefinite perspective hs (V. P.)^ should pass 
through c, and the indefinite perspective e (V. P.)^ 
should also pass through c^. 

The perspective could have been drawn without 
the use of the fines s^c^, s^b^, etc., by using only 
the five vanishing points. The intersection, of 
the indefinite perspectives ai (V. P.)^ g (V. P.)^, 
in as gives one point in the perspective, a^ (V. P.)^ 
bi (V. P.)^, in bs gives a second, bs (V. P.)^ Ci 
(V. P.)^ in Cs gives a third point, and Ci (V. P.)^ 
e (V. P.)^ should also give c^. 

XIII. The best method and the one that takes 
the least number of lines is the method by the 
vanishing point of rays, and will now be described. 
It consists of finding the vanishing point of the 
horizontal projections of the rays of light and the 
vanishing point of the fight rays themselves. 
The orthographic projection of the shadow in the 
horizontal plane is entirely ignored and has no 
use in this method. The horizontal projection 
of the rays of fight make 45° with the X X axis, 
and the vanishing point of these is found in 
(V. P.)^ as already described. The vanishing 



32 ELEMENTARY COURSE IN PERSPECTIVE 

point of the rays themselves is found by passing 
through s^ and s"" hnes parallel to the horizontal 
and vertical projections of the rays, which accord- 
ing to conventionality are 45° and 315° respec- 
tively, with the X X axis, and where it intersects 
the vertical or picture plane is the vanishing 
point of the rays themselves. This vanishing 
point is found at (V. P.)^ The perspective will 
now be drawn. The indefinite perspective of the 
horizontal projection of the ray through a is a^ 
(V. P.)*, and the indefinite perspective of the 
ray itself is a (V. P.)^ These two hnes meet in 
as, and this is where the ray itself intersects the 
horizontal plane, and is therefore the perspective 
of the shadow of a in the horizontal plane. The 
indefinite perspective of the horizontal projection 
of the ray through h is hi (V. P.)^ and the in- 
definite perspective of the ray itself is h (V. P.)^ 
These two hnes intersect in bs, and this is where 
the ray itself intersects the horizontal plane and 
is therefore the perspective of the shadow of h in 
the horizontal plane. In the same manner the 
shadow of all points of the object is found. To 
complete the shadow join g, as', as, hs', etc., for all 
points of the shadow found. Thus we get as our 
complete shadow gaJbsCsC of the frustum and the 
shaded portion is the visible part of the perspec- 
tive of the shadow. 




t^=^^zz:^S 









u<i^- 



Uv-p.)"* 










N(V.P.)' 



::.'>: 



Fig. 6. 



ll 



PERSPECTIVE OF SHADOWS 33 

XIV. Fig. 7 shows the reproduction of the per- 
spective of a cube and a parallelopiped block, 
resting on one edge of the cube, carefully described 
Fig. 4. In addition to this is shown the per- 
spective of the shadow of the object by the method 
of the vanishing point of rays, and the description 
will be confined to the details of this method. 
(V. P.)^ is the vanishing point of the horizontal 
projection of the rays and (V. P.)^ is the vanish- 
ing point of the rays themselves. The perspec- 
tive of the horizontal projection of the ray through 
h is 6i (V. P.)^, and the perspective of the ray 
itself is h (V. P.)^, these rays meet in hs and is the 
perspective of the shadow of b, and hj)s is the per- 
spective of the shadow of hb^. 

The perspective of the shadow of c is found at 
Cs by the same method that bs was found, and 
bs Cs is the perspective of the shadow of be. The 
line cd is parallel to the horizontal plane, and 
hence the vanishing point of its shadow must be 
the same as the vanishing point of the Une. 
Therefore c^ (V. P.)^ is the indefinite perspective 
of the shadow of dc. The perspective of the 
shadow of dd^ is found exactly as bb^, and its 
indefinite perspective is d^ (V. P.)^ This gives 
the complete shadow of the cube, but on account 
of the parallelopiped resting on the cube there 
is a certain portion of the cube that does not cast 



34 ELEMENTARY COURSE IN PERSPECTIVE 

a shadow, and that portion of the cube that does 
not cast a shadow, the parallelopiped will cast 
a shadow on it. 

The horizontal projection of the point n is rii, 
found by passing a vertical plane through fn and 
(V. P.)^ intersecting the horizontal plane in the 
line / (V. P.)^ This same vertical plane inter- 
sects the face dh of the cube in the Hne I (V. P.)^ 
I being the point where the edges fn and ad inter- 
sect. From n draw a line perpendicular to X X 
axis and intersecting line / (V. P.)^ and the hori- 
zontal plane in Ui. This same perpendicular 
intersects the line I (V. P.)^ and the horizontal 
plane dh of the cube in the point UiK The hori- 
zontal projection of the ray through n is n^ 
(V. P.)^ and the same projection of the ray on 
the plane dh of the cube is n^^ (V. P.)^ The per- 
spective of the ray itself is n (V. P.)*, and inter- 
sects rii (V. P.)^ and rii^ (V. P.)^ in n^ and n/, 
respectively, which are the perspectives of the 
shadow of the point n on the horizontal plane of 
the base of the cube and the plane dh of the face 
of the cube respectively. The perspectives of 
the shadow of / and I are / and I themselves, 
since they are in planes that the perspective is in. 
The perspective of the shadow of the Hne fn 
would be fn^ if there were no cube to cast a 
shadow on, but since that is the case, the shadow 



PERSPECTIVE OF SHADOWS 35 

falls on the horizontal plane in two different parts 
and on two faces of the cube, namely, ad^ and d 6. 
The first part of the line fn considered is /Z and 
the perspective of its shadow is that portion of fUs 
between / and the point r^ where it intersects 
the perspective d^ (V. P.)^ of the shadow of the 
edge ddi, and that part lying in the face adi of the 
cube. This shadow lying on the cube is found by 
extending d^ (V.P.)^ to /r„ and the former inter- 
sects the latter in hs. Then hs hes on the face ad^ 
produced and hs is a shadow of some point in 
fn. Therefore the portion of the shadow of fn 
on the indefinite plane ad^ is Ihs and it intersects 
ddi in r/. The shadow lying on the cube is 
hence Zr/. The point casting the shadow r^^ 
also casts the shadow r„ for the perspective of 
the ray through r] is r], (V. P.)^ and it passes 
through Vs. Hence the perspective of the shadow 
of fl is frs and r] I. The second portion of the fine 
fn is In, and the perspective of the shadow lies on 
the face dh of the cube and that portion of the 
fine fns between n^ the perspective of the shadow 
of n and the intersection Os of the perspective 
fns and c^ (V. P.)^ The part on the face dh is 
found by joining Zn/ and that portion of the 
fine Zo/ included between the sides ad and dc 
in the perspective of the shadow of the line 
fn on the face dh. The remainder of the line 



36 ELEMENTARY COURSE IN PERSPECTIVE 

0/ n/ has the perspective of its shadow in the 
hne OsUs on the horizontal plane. The perspec- 
tive of the shadow of the line fn is now com- 
pleted, and is represented by /r„ r/Z; Zo/; and 
OsU,. The line shown joining r^ and r/, also 0/ 
and Os are in the perspective of the same ray of 
light, it being the point where the perspective is 
just leaving one surface and entering upon another. 
The next point is to determine what Unes in the 
end gnik cast shadows on the horizontal plane 
of the base of the cube. This is determined 
by finding which of the four lines have their 
shadow the farthest to the right. There can 
be only two of them that cast a shadow and 
these are found to be ni and ik. The perpendicu- 
lars from I and k are found the same as for n, 
and in the case of k is shown as ki. The perspec- 
tives of the shadows of i and k are found exactly 
as for Uy and are shown at is and ks. The per- 
spectives of the shadows of ni and ik are hence 
Us is and i^ k^. The next element of the parallelo- 
piped that casts a shadow is mk. The shadow 
of k has been found and is k^, while that of m is 
wis, and found in the same way as k^. The shadow 
of mk is hence m^ k^. This shadow falls outside of 
the shadow of d, hence this hne m, k^ is the limit 
of the shadow in this direction. The shadow of 
the end femm next found, and since this plane 



PERSPECTIVE OF SHADOWS 37 

is parallel to the plane gkin, the lines casting the 
shadow will be the reverse of those casting the 
shadow in plane gkin. In gki n, ni and ik cast 
shadows while in plane fern, fe and em must 
be the hnes casting shadows. The shadow of 
these latter two Hnes are found in exactly the 
same manner as the former two lines. The entire 
perspective of the shadow and of the cube and 
parallelopiped have been found and the visible 
portions of this perspective have been cross- 
hatched. The outUne of the shadow has been 
lettered at almost every angle, and those points 
that have been lettered will be mentioned to show 
clearly the outline of the perspective. They are 
as follows: — 61, 6„ c^, o„ n^, is, K, m„ e^ (not 
shown), /, r„ d^, c^ (not shown), and 61 in the hori- 
zontal plane, while on the cube they are r/, 
I, 0/, d, r/. 

XV. Fig. 8 contains a given oblique plane W 
whose traces are H. W. and V. W., i.e., horizon- 
tal and vertical traces * of the plane W respec- 
tively, and the perspective of the frustum of the 
pyramid the same as shown in Fig. 6 being an 
object in angular perspective. 

The horizontal and vertical projections of the 

* By traces of a plane are meant the lines of intersection of 
the plane with the planes of projection, i.e., in perspective 
horizontal and picture plane. 



38 ELEMENTARY COURSE IN PERSPECTIVE 

frustum are shown as a^Vd^e^g^ etc., and 
aVc'^d^'e'^g'' etc., respectively, and its perspective 
by ahcdeg, etc. The method of drawing this 
perspective has been described in previous arti- 
cles under angular perspective, and the vanishing 
points of the horizontal lines are shown as (V. P.)^ 
and (V. P.)^ Under previous articles on the per- 
spective of shadows it was shown how to find the 
vanishing point of the horizontal projection of 
rays and also the vanishing point of the rays 
themselves, and their vanishing points in Fig. 8 
are (V. P.)^ and (V. P.)^ respectively. 

The problem in Fig. 8 is to find the perspec- 
tive of the shadow of this frustum ABC D EG, 
etc., on the given oblique plane W, It can be 
found in two ways, 1st, by finding the orthographic 
projections of the shadow on the plane W and 
then find the perspective of the shadow from 
these projections. 2d, by finding the perspec- 
tive of the shadow on the oblique plane W direct 
without the use of the horizontal and vertical 
projections of the shadow, which will be the 
method now described and illustrated in this 
figure. 

Since the conventional direction of rays is 
such that their horizontal and vertical projec- 
tions make 45° and 315° with the XX axis respec- 
tively, we have projections of these lines as c* 



PERSPECTIVE OF SHADOWS 39 

(h. i.)S c^w"; ^ (h. i.)^ e^ w\ etc., respectively. 
Imagine vertical projecting planes V , F, Z, etc., 
passed through these lines, and their horizontal 
and vertical traces shown by H. U., V. U. ; H. Y., 
V. Y.; H. Z., V. Z., etc. Since the rays of light 
are parallel, these planes [/, F, Z, etc., are paral- 
lel. These horizontal traces H. U., H. Y., H. Z., 
etc., intersect the horizontal trace H. W. of the 
plane W in (h. i.)^; (h. i.)^; (h. i.)^, etc., respec- 
tively, and the vertical traces V. U., V. Y., V. Z., 
etc., of these same planes intersect that of W in 
(v. i.) *; (v. i.) ^; (v. i.) ^ etc., respectively. These 
points hence lie in the horizontal and vertical 
planes of projection respectively. These planes 
C/, F, Z, etc., if produced far enough will inter- 
sect the given obUque plane W in straight lines 
which in space pass between the points (h. i.) ^, 
(v. i.)^; (h. i.) ^ (v. i.) ^; (h. i.) ^ (v. i.) % etc., and 
whose projections are & (h. i.) ^, (v. i.) ^ A;J; e^ 
(h. i.) ^ (v. i.) 2 ml\ g^ (h. i.) ^ (v. i.) ' ol etc. 
These lines of intersection with plane W are all 
parallel and hence have the same vanishing 
point. To find the vanishing point of the inter- 
secting lines between the vertical projecting 
planes and the plane W pass a line through the 
station parallel to these lines, and where it inter- 
sects the picture plane will be the vanishing point 
of this system of parallel hnes. Therefore through 



40 ELEMENTARY COURSE IN PERSPECTIVE 

the horizontal projection of the station s^ draw a 
hne parallel to c^ (h. i.) '; e^ (h. i.) '; g^ (h. i.) ^ 
etc., intersecting the XX axis in nj. Now 
through the vertical projection of the station s"" 
draw a line parallel to (v. i.) ^ kl; (v. i.) ^ ml; 
(v. i.) ^ ol, etc., and where it intersects a perpen- 
dicular to the X X axis through nj is where this 
line through the station intersects the picture 
plane, as (V. P.) ^ is the vanishing point of the 
above system of lines. The lines of intersection 
between the planes U, Y, Z, etc., and the given 
obhque plane W intersect the picture plane in 
points (v. i.) ^; (v. i.) ^; (v. i.) ^, etc., and the 
perspective of these points are the points them- 
selves. The indefinite perspectives of these inter- 
secting lines are therefore (v. i.) ^ (V. P.) ^; (v. i.) ^ 
(V. P.) '] (v. i.) ' (V. P.) ^ etc., and of the rays 
themselves are c (V. P.) '; e{Y. F.Y; g(V. P.)^ 
etc. Since the rays of light themselves and these 
intersecting lines lie in the same planes they will 
intersect and their points of intersection will 
be in the plane W and hence the shadows of some 
points of the object. Therefore, the intersec- 
tions of these two systems of indefinite perspec- 
tives will be the perspectives of the shadows of 
the corners of the frustum of the pyramid on 
the given oblique plane W. 
We will now find the perspective of the shadow 



PERSPECTIVE OF SHADOWS 41 

of the point C on the given oblique plane W. 
The horizontal projection of the ray of Hght 
through C is c^ (v. i.) \ and the traces of the ver- 
tical projecting plane U containing this ray of 
light are H. U. and V. U. respectively. The 
horizontal and vertical projections of the line of 
intersection between planes U, and W are c^ 
(h. i.) \ (v. i.) ^ kl and the indefinite perspective 
of this hne of intersection is (v. i.) ^ (V. P.) ^ 
On this indefinite perspective is the perspective 
of the shadow of the point C on the plane W. 
Also the shadow of the point C is on the ray 
through C and is where the ray intersects or 
pierces the plane W, The perspective of C is c 
and the indefinite perspective of the ray through 
C is c (V. P.)* which contains the perspective of 
the shadow of C on the plane W, Since the per- 
spective of the shadow of C on the plane W is 
on both systems of indefinite perspectives it 
must be at the intersection of these two systems. 
Therefore the intersection cT of the two indefinite 
perspectives (v. i.)^ (V. P.)^ and c (V. P.)^ will 
locate the perspective of the shadow of C on the 
given plane W. By the same method we can 
get the perspective of all the points of the frus- 
tum of the pyramid that cast a shadow on the 
plane W. 

In order to determine what points cast the 



42 ELEMENTARY COURSE IN PERSPECTIVE 

shadow on the obhque plane IF, we can find 
the perspective of the shadow of all corners of 
the frustum, and those that fall within the ex- 
treme or limiting points of the shadow will be 
obliterated by these extreme points. By a careful 
observation one is able to discover which points 
fall within the shadow boundary without the 
labor of finding the exact perspective of the 
shadow of each point on the plane W, 

As a check on the perspective of the shadow of 
the frustum on the plane W, we can find the ver- 
tical projections of the points where the rays of 
hght intersect the plane W. By finding the dis- 
tance the points are below the horizontal plane of 
projection we can find the indefinite perspectives 
of horizontal lines containing these intersections 
and lying in vertical projecting planes containing 
the rays of light. These rays themselves and these 
horizontal fines intersect the plane W in the same 
point and therefore intersect each other at that 
point. The perspectives of these points of inter- 
section of the rays and the horizontal lines give 
also the perspectives of the shadows of the points 
of the frustum on the obhque plane W. 

Two of the checks have been shown in Fig. 
8, but here will be described only one of them 
as follows: Draw the vertical projection of the 
horizontal plane showing how far this plane is 



PERPSECTIVE OF SHADOWS 43 

above the frustum of the pyramid. This line 
corresponds to the XX axis shown above and 
is lettered XjXi. Produce kl (h. i.)^ until it 
intersects the axis XjXi in the point kl^y and 
through this point draw kl^w"" parallel to kl 
(v. i.)^ This line is then the vertical projection 
of the line of intersection of the planes U and W, 
Now through the vertical projection c" of the cor- 
ner C of the frustum, draw the vertical projec- 
tion c"w'" of the ray, and where it intersects the 
line kliW"" in the point w'" is the vertical projec- 
tion of the intersection of the ray through C 
with the plane W. This projection shows the 
true distance that the point W is below the hori- 
zontal plane, and a horizontal Hne passed through 
this point W intersects the picture plane at the 
same distance w'" lli below the horizontal plane. 
We next lay off this distance w"" ll^ below the X X 
axis and on the vertical trace V. U. of the plane 
U, since this horizontal line through the point W 
lies in the plane U, and the point cj' is located. 
Since this horizontal line lies in the plane Uj its 
vanishing point will be (V. P.)^ and the indefinite 
perspective of this Hne is cj' (V. P.)^ and should 
intersect the perspective of the ray c (V. P.)* in 
the same point c^ that the perspective of the 
line of intersection (v. i.)^ (V. P.)^ does the per- 
spective of this ray. It is seen from Fig. 8 



44 ELEMENTARY COURSE IN PERSPECTIVE 

that this point c^ does coincide with the first 
position of c^ and that the check proves that 
there is no error in the first method of obtaining 
the perspective c^ of the shadow of the point C. 
In Uke manner the drawing shows the check on 
the perspective e^ of the shadow of the point E 
and also demonstrates that there is no error in 
locating this point. All the points of the perspec- 
tive of the shadow could be checked by follow- 
ing the same method as just described, but what 
has been said will be sufficient to understand the 
principles underlying this check. 

The lines casting the shadow will now be given 
in the order of sequence of connecting points as 
EC, CB, BA, AG, GF, FE, and the perspectives 
of the shadows of the ends of these lines, in the 
same order of sequence on the obhque plane W, 
are e^, c^, 6^, <, gr^, /^. After these points are 
connected in the proper order it is seen that part 
of the shadow is out of sight, being behind the 
perspective of the object, and that the outline of 
this part is in dotted fines. The portion of the 
shadow that is visible is cross-hatched and its 
corners are /^, e^, c^, 6^, a^, while the invisible 
corner is g'^. Thus the complete perspective of 
the shadow of the frustum of a pyramid on an 
oblique plane is shown. 



PERSPECTIVE OF SHADOWS 45 

PROBLEMS IN PERSPECTIVE OF SHADOWS 

XVI. 1. Find the perspective of the shadow 
cast by the object as shown in Fig. 2. 

2. Find the perspective of the shadow cast by 
the object as shown in Fig. 3. 

3. Find the perspective of the shadow of the 
object as shown in Fig. 5. 

4. Find the perspective of the shadow of the 
object as given in question 18, page 14. 

5. Find the perspective of the shadow cast 
by the object as given in question 25, page 15. 

6. Find the perspective of the shadow cast 
by the object in question 26, page 15. 

7. Find the perspective of the shadow cast 
by the object in question 27, page 15. 

8. Find the perspective of the shadow of 
the object as given in question 29, page 16. 

9. Find the perspective of the shadow of the 
object as given in question 2, page 28. 

10. Find the perspective of the shadow of the 
object as given in question 5, page 28. 

11. Find the perspective of the shadow of the 
object as given in question 6, page 28. 

12. Find the perspective of the shadow of the 
object as given in question 7, page 28. 

13. Find the perspective of the shadow of the 
object as given in question 1, page 21. 



46 ELEMENTARY COURSE IN PERSPECTIVE 

14. Find the perspective of the shadow of the 
object as given in question 4, page 21. 

15. Find the perspective of the shadow of the 
object as given in question 6, page 21. 

16. Find the perspective of the shadow of the 
object as given in question 16, page 23. 

17. Find the perspective of the shadow of the 
object as given in question 22, page 24. 

18. Find the perspective of the shadow of the 
object as given in question 25, page 24. 

19. Find the perspective of the shadow of the 
object as given in question 27, page 24. 

20. Find the perspective of the shadow of the 
object as given in question 29, page 24. 

21. Find the perspective of the shadow on an 
obhque plane of the frustum of a right circular 
cone when its base is parallel to the horizontal 
plane. 



Fig. 7 




\ 

\ 



I 



PART II 
PERSPECTIVE BY THE METHOD OF SCALE 



Chapter VI 
PERSPECTIVE BY THE METHOD OF SCALE 

XVII. The first article on perspective was writ- 
ten by using the orthographic projections of the 
object and the rays to obtain the perspective of 
the same. A pictorial illustration of this method 
is shown in Fig. 1, and the actual drawing of this 
method for that particular object is shown in 
Fig. 2. 

The second article treats of the method by 
scale, and Fig. 10 shows this method for the same 
object as well as for the same position as was 
used in Fig. 2. 

If the first method is used you have to place 
the plan, not necessarily draw it, upon the draw- 
ing-board. This as a rule requires a wide draw- 
ing-board and a long straight-edge to draw the 
horizontal projections of the rays. In this 
method you get both the perspective of points 
and lines of the object. This method usually 
requires fewer fines and less time to draw the 
perspective, after the plan and elevations are 
correctly placed, than the second method does. 

49 



50 ELEMENTARY COURSE IN PERSPECTIVE 

In the second method, as well as in the first 
method, the projections of the object are on 
different pieces of paper from those upon which 
the perspective is drawn. In the first method 
they are usually tacked to the drawing-board in 
their proper position for drawing the perspective. 
In the second method they simply he loosely 
upon the drawing-board to be used only for 
scahng or computing distances. In this second 
method, the method by scale, the drawing-board 
does not have to be as large, and hence does not 
take so much room to draw it. In this method 
we very seldom draw the perspective of a fine of 
the object, unless we do it as a check, it always 
being to find the perspective of a point. The 
perspective of a point being found by the inter- 
section of the perspective of two horizontal fines, 
lying in the same plane as the given point, and 
making 45° with the picture plane. This shows 
that for the perspective of each point you must 
have the perspective of at least two lines, and the 
point where each one of these fines intersects the 
picture plane must be scaled in order to get their 
perspective. If you wish a check on this work 
you must have a third horizontal line in the same 
plane as the other two lines, and the point where 
this intersects the picture plane must also be 
located by scafing before its perspective can be 



PERSPECTIVE BY THE METHOD OF SCALE 51 

drawn. This third or check line is usually per- 
pendicular to the picture plane, and hence its 
vanishing point is in the center of picture. You 
can now see that the advantage of the second 
method is that it takes a smaller board than the 
first method, especially in width, while the length 
in each method is about the same. The disad- 
vantage is that there is a great deal of scaHng 
and the perspective of a great many construction 
lines, most of them being horizontal 45° lines. 



Chapter VII 
PARALLEL PERSPECTIVE 

XVIII. Figure 9 gives the projections of a right 
rectangular pyramid, so placed as to make a 
parallel perspective. It is the same size and 
placed at the same distances from the picture 
plane and horizon plane as in Fig. 2, in Part I. 
The pyramid has a base 23 feet below the horizon 
plane and is 7 feet by 12 feet. It has an altitude 
of 10 feet and the vertex is 24 feet behind the 
picture plane. The station is 15 feet in front of 
the picture plane, and is to the left of the center 
of the object 16 feet. In this drawing, as well as 
all orthographic projections of objects in this 
part, the XX axis will be the intersection of 
the picture plane and the horizon plane. This 
makes the horizontal projection of the picture 
plane and the vertical projection of the horizon 
coincide with each other, and with the X X axis. 
In the first article the horizontal plane was parallel 
to the horizon plane, while in this article it coin- 
cides with the horizon plane. 

This figure also shows the horizontal projection 

52 



PARALLEL PERSPECTIVE 



53 



of horizontal, 45° construction lines for the points 
A J By C, and D. The horizontal projection for 
the 45° construction lines for A are a^ a^ and 
A third construction line for a check 



a^a[ 



r.-7-^-H 




^ \ ^ 

Jt ,<y^P^ at/ / ivH-Pfcfure yfjVM \ Plane. \ \\\ y 

it \ (\-^ — I^ ^/^4* ^ — >^i^ 



V / 






Ijy/?/ 



^i 



Plan 



a^ 



Elevat/on 

Scale r=20ft. 
Fig. 9. 

would be o!" a\, perpendicular to the picture 
plane. In like manner we have for the point B, 
V'hl] 6^3^ and ^Hf. 

In the same way for points C and B, The 
vanishing points of the 45° Hnes, the hues per- 



54 ELEMENTARY COURSE IN PERSPECTIVE 

pendicular to the picture plane, and the Hne A B 
are shown at (V. P.)f; (V. P.)?; (V. P.)', and 
(V. F.y, respectively. 

In the case of the vanishing point of 45° lines 
we have formed an isosceles triangle, one leg 
being perpendicular to the picture plane, from 
the station, and intersecting it in the center of the 
picture. The other leg is drawn from the center 
of picture, and lying in the picture plane until it 
intersects the 45° horizontal lines through the 
station. Figure 9 shows one right-angle triangle 
in horiozntal projection as s^, (V. P.)^ (V. P.)^, 
and the other as s\ (V. P.)^ (V. P.)?. These 
two triangles are equal and hence (V. P.)'' (V. P.)? 
equals (V. P.)^ (V. P.)?, and since these two tri- 
angles are isosceles, s^ (V. P.)^ equals (V. P.)^ 
(V. P.)f equals (V. P.)^ (V. P.)?, and hence the 
vanishing points of 45° horizontal lines are as 
far to the left or right of the center of the picture 
as the station is in front of the picture plane. 

To draw the perspective a, h, c, d in Fig. 10 of 
the object ABC D by the use of Fig. 9 will now 
be described. First draw the horizon and 23 
feet below it draw the line representing the inter- 
section of the plane of the base of the pyramid 
with the picture plane. These two lines give the 
extreme vertical hmits of the drawing. The 
point B is the farthest visible point of the object 



PARALLEL PERSPECTIVE 55 

behind the picture plane as shown in Fig. 9, and 
the 45° lines from this point will intersect the 
picture plane farther to the left and right than 
any other 45° hues from a visible point of the 
object. B is 30 feet back of the picture plane, 
and hence where h^h^ intersects the picture 
plane is 30 feet to the left of 5, and h^ h^ inter- 
sects 30 feet to the right of B, The greatest 
width of the picture is now found to be 60 feet, 
while the greatest height is 23 feet. With these 
our perspective can be located in a proper posi- 
tion on our sheet. 

On the base hne, which is the intersection of 
the base plane of the object with the picture 
plane, in Fig. 10, locate 6J and 6J 60 feet apart. 
The vertical projection of B is h"" 30 feet to the 
right of h^. The station is 16 feet to the left of 
A and hence 12.5 feet to the left of B, which 
makes its projection s^ in the base line 12.5 feet 
to the left of h" or 17.5 feet to the right of h'^. The 
vertical projection of the station is in the horizon 
and directly above sj", as shown at s"", and is also 
the vanishing point of perpendicular horizontal 
lines, and is lettered V. P. The vanishing points 
of the 45° hnes are 15 feet to the left and right of 
s", and are shown as (V. P.)/ and (V. P.),. The 
point sj' is the initial point for measuring all 
distances in the base Hne. The perpendicular 



56 ELEMENTARY COURSE IN PERSPECTIVE 

and 45° lines through B intersect the base Hne in 
h^^j hi, and hi, which are 12.5 feet to the right, 
17.5 feet to the left, and 42.5 feet to the right of 
s^. The indefinite perspectives of these lines are 
6" V. P.; hi (V. P.),; and hi (V. P.),, respectively, 
and they all meet in the point h, which is the 
perspective of B. The three lines through the 
point C intersect the base line in h"", cl, cl, which 
are 12.5 feet to the right, 5.5 feet to the left, and 
30.5 feet to the right of s^. The indefinite per- 
spective of these fines intersect in c the perspec- 
tive of C. The three lines through D intersect 
the base line in d", dl, dl, which are 19.5, 1.5, 
37.5 feet to the right of s^. Their indefinite per- 
spectives intersect in d, which is the perspective 
of i). 

The horizontal plane passing through the ver- 
tex intersects the picture plane 10 feet above the 
base fine or 13 feet below the horizon. The con- 
struction fines through A lie in this plane and 
intersect the picture plane. Two of these fines 
are sufficient to get the perspective of A, the 
third line serving as a check. These fines inter- 
sect the picture plane in the points a", aj, al, 
which are 16 feet to the right, 8 feet to the left, 
and 40 feet to the right of s^ and 10 feet above it. 
These indefinite perspectives intersect in the 
point a which is in the perspective of A. 



PARALLEL PERSPECTIVE 57 

The perspective of the pyramid is now found 
by joining the points a, h, c, d in proper sequence, 
and we have the parallel perspective of the right 
rectangular pyramid ABC D, As a check c d 
should be parallel to the horizon line, since CD 
is a horizontal hue and parallel to the picture 
plane. A check on a 6 is to find the vanishing 
point of the Une A 5 as described under Fig. 2. 
In Fig. 10 we have computed the proper distances 
to the left of the vertical projection s"" of the 
station and below the horizon in order to scale 
and locate this vanishing point of A 5 as shown 
in (V. F.)\ It is 8.75 feet to the left of s", and 
25 feet below the horizon, and the Une (V. P.)^ b 
produced should contain a as a check. 

XIX. Figure 11 shows the projection of a build- 
ing in which one end is parallel to the picture 
plane. The dimensions of the building are clearly 
shown in its projections, but the location of the 
building and the station with reference to the 
picture plane and the horizon plane are not 
shown in the drawings and will be given here as 
follows: near end is 16 feet behind the picture 
plane, the base of the building is 24.5 feet below 
the horizon plane, the station is in front of the 
picture plane 111 feet, and to the right of the center 
of the near end of the building 190 feet. 

Figure 12 is the prespective of Fig. 11 by the 



'5'8 ELEMENTARY COURSE IN PERSPECTIVE 

method of scale, and is a more complicated case 
of parallel perspective than in Fig. 10. It is the 
same building and in the same location and posi- 
tion as in Fig. 3, Part I. You are thus able to 
compare the two methods and judge for your- 
self which method you consider the easier and 
quicker. 

Let the horizon first be drawn, and 24.5 feet 
below it the line representing the intersection of 
the plane of the base and the picture plane. On 
this last hne is where all 45° and perpendicular 
lines of the base intersect the picture plane. 

On the horizon locate the center of picture or 
vertical projections s" of the station, which is 
also the vanishing point V. P. for perpendicular 
lines. The vanishing points of the 45° hues are 
on the horizon, since they are horizontal lines, 
and as far to the left and right of the vertical pro- 
jection of the station s" as the station is in front 
of the picture plane, as (V. P.)/ and (V. P.)^, 
respectively. Now lay off on the horizon 190 
feet to the left, and we have the point m". If the 
end view of the building were projected upon the 
picture plane, we would have the following out- 
line a" 0'" p" r'"c''c1a\', ayi being 24.5 feet below 
m", and is in the intersection of the plane of the 
base of the building and the picture plane. The 
lines a"" a^ and c"" c^ are each perpendicular to 










-~~sJ 



\ 



-4'— 



A" 



Scale 1''= 6 ft. 
Fig. 10. 



<.V.R\ 



lV.R)r 




^ ^ 



PARALLEL PERSPECTIVE 59 

a"^c'"^j and equal distances from nf being one 
half the width of the building. The indefinite 
perspectives of the lines from the points a"^ aj", 
c'^j c", etc., in the picture plane to infinity are 
drawn from a" to V. P.; a^" to V. P., etc., as de- 
scribed in the first part under Fig. 3. The defi- 
nite perspective of the lines a" V. P.; a^ V. P., 
etc., are now found by the indefinite perspectives 
of the 45° lines through the points Aj Ai, etc. 
The station projected from s" onto the base line 
is shown at s^, and this is the place from which 
all measurements are taken along the base fine. 
The 45° lines through A^ intersect the picture 
plane 221.5 and 189.5 feet to the left of s^, and 
are shown as aj and ttg respectively. They 
should intersect each other on the indefinite per- 
spective a^ V. P. as at a^ which is the perspective 
of Ai. The 45° fines through Ci intersect the 
picture plane 190.5 and 158.5 feet to the left of 
s^, and are shown as c^ and c^ respectively. 
These lines vanish in (V. P.);, and (V. P.)/ re- 
spectively. They should intersect each other in 
the indefinite perspective c^ V. P. at Ci, which is 
the perspective of Ci. The perspectives of the 
points A and C are shown at a and c, and are 
obtained in the same way as a^ and Ci, except in 
a plane parallel to the plane of the base and 30.5 
feet above it. The perspective of P is found by 



60 ELEMENTARY COURSE IN PERSPECTIVE 

finding the perspective of three fines passing 
through P, one being perpendicular and the 
other two making 45° with the picture plane. 
The perspective of the point where they meet is 
the perspective p of the point P. The perpen- 
dicular line and the 45° lines intersect the picture 
planes 190, 212, 168 feet to the left of < respect- 
ively, and 47 feet above it as shown at p'", pl, p^. 
These three lines have their vanishing points in 
V. P.; (V. P.)r; (V. P.)/, and where they meet is 
the perspective p of the point P. The perspec- 
tive of Pi is on the indefinite perspective of p'' V. P. 
and in the plane of the near end of the building. 
One 45° fine through Pi intersects the picture 
plane 174 feet to the left of s"^ and 47 feet above 
it as at p^. The vanishing point of this fine is 
(V. P.)/ and its indefinite perspective is p^ (V. P.)/ 
intersecting p'^Y. P. in pi the perspective of Pi. 
The perspective of the other 45° line is not shown 
and would serve only as a check in the perspec- 
tive pi. Join a pi and c pi and we have the per- 
spectives of A Pi and C Pi. On these perspectives 
are the perspectives of and R respectively, and 
hence the perspective oi R. P is a horizontal 
line and parallel to the picture plane, hence its 
perspective is parallel to the horizon. This fine 
is 6 feet below Pi therefore its vertical projection 
d" r" wifi be 6 feet below the vertical projection 



PARALLEL PERSPECTIVE 61 

p" of the point P. The middle point of o'"r'' is 
h"", which is the projection of B. The perspective 
of 5 is on a perpendicular through pt and where 
the indefinite perspective ^'^ V. P. intersects it is the 
perspective h of the point B. The perspective oiOR 
passes through h and is parallel to the horizon, and 
intersects a pi and c pi in o and r respectively. 
As a check the indefinite perspective o^'Y. P. and 
r" V. P. should pass through o and r respectively. 
We now have ao, or, and c r as the perspectives 
oi AO, R, and C R. The perspectives of P 
and RP are now drawn and are shown as op 
and rp. As a check on these last two lines we 
can compute the distances of their vanishing 
points and locate them by scale. The vanishing 
point oi R P will be 111 feet above (V. P.)z and 
that of P, 111 feet above (V. P.).. By con- 
necting the points in proper sequence we have 
the perspective a^aopreci of the end A^AOP 
RCC„ 

By the same method the perspective of Fi can 
be found as shown at/i. The construction lines 
intersect the picture plane 164.5; 206.5 (not 
shown) and 122.5 feet to the left of <. The 
first intersection is shown at f^ and the third at 
f^, and the second intersection would be used 
only as a check. By following the same methods 
you can locate the perspectives of all the corners 



62 ELEMENTARY COURSE IN PERSPECTIVE 

of the building, and we have finally the perspec- 
tive aiaopdflkj, etc. of the object AiAOP 
DFLKJ,etc, 

The above description with the aid of the addi- 
tional construction lines should be enough to give 
a thorough understanding of the perspective in 
Fig. 12 and of the object shown in Fig. 11. 

PROBLEM IN PARALLEL PERSPECTIVE 

XX. Find the perspective of all the problems 
under parallel perspective in Part I, but solve them 
all by the method of scale. 



Hor/zon {V.R)r 




■iZ^ 



H 



End Elevation 
Scale 1'*= 24 ft. 
Fig. 11. 



W. 






^ 



a/\cy__,_ e^^l 



'^''"-ef^ff-'-'-"Sf''''l^"cl''--- 



Scale 1"= 30 ft. 
FiQ. 12. 



Chapter VIII 
ANGULAR PERSPECTIVE 

XXI. Figure 13 shows the working drawings of 
a cube and a parallelepiped block resting upon one 
edge of the cube. It occupies the same position 
as in Fig. 4. The faces of the cube being 30° 
and 60° with the picture plane, and the near 
edge of the cube 4 feet behind the picture plane. 
The base of the cube is 11.6 feet below the horizon 
plane. The station is 4.4 feet to the right of the 
near corner of the cube and 21.2 feet in front of 
the picture plane. This makes the vanishing 
point of 45° hues 21.2 feet to the left and right 
of the center of picture. Figure 13 also shows 
such computed distances as will enable you to 
find the distance to the left or right of the center 
of picture, and the distance above the base plane 
of the points where the perpendicular and 45° 
lines intersect the picture plane. 

Figure 14 is the perspective of the object as 
shown in Fig. 13. The horizon is drawn, and 
11.6 feet below it is shown the intersection of the 
plane of the base of the cube and the picture 

63 



64 ELEMENTARY COURSE IN PERSPECTIVE 

plane. The vertical projection of the station is 
shown at s", and is also the vanishing point of 
lines perpendicular to the picture plane, as well as 
the center of picture. The projection of this 
point on the base Une is shown at s^, from which 
measurements are taken to the left or right. 

The perspective of the point Bi will first be 
located. A perpendicular line through B^ inter- 
sects the picture 4.4 feet to the left of the station, 
and its vertical projection is 4.4 feet to the left 
of si as shown in 6J'. The indefinite perspective 
of this perpendicular line is fcJ'V. P. The two 
45° fines intersect the base fine in h'^ and b^ 
(4.4 d= 4.0 = 8.4 and 0.4), 8.4 and 0.4 feet respect- 
ively to the left of s'^. The indefinite perspec- 
tives of these 45° fines are h^ (V. P.), and h^ 
(V. P.)/ These three perspectives should meet in 
a point, and where they meet is hi the perspective 
of Bi. The perspective of A^ is found by the 
same class of lines, and they intersect the picture 
plane in a^, a^, and aj, each being 11.33, 19.33, 
3.33 to the left of s^. The perspective of C is in 
a plane 8 feet higher than the base plane. In 
this plane we use perpendicular and 45° fines as 
in the base plane, and they intersect the picture 
plane in c^^, c'^, cl, which are 0.4, 11.33 feet to the 
left and 10.93 feet to the right of < and 8 feet 
above the base plane. These three indefinite per- 



ANGULAR PERSPECTIVE 65 

spectives cl V. P.; < (V. F.)/, cl (V. P.)/ meet in 
the same point c, which is the perspective of C. 
In Hke manner the perspective of all the corners 
of the cube can be found and connected in the 
proper sequence, thus completing the perspective 
of the cube A^ Bi B C D. Another check is to 
find the vanishing points of the systems of lines 
A B, etc., and B C etc. They can be computed 
and are in the horizon to the left and right of s" 
36.72, 12.24 feet, and are shown as (V. P.)'; 
(V. P.)^ 

The perspective of the parallelopiped block will 
now be drawn. One edge of this block rests in 
the base plane, and we will find the perspective 
of the point F of this edge. The three construc- 
tion hues that we wish to use intersect the picture 
plane on the base Hne in the points /"", f^, f^, and 
are 19.09, 33.65, 4.53 feet to the left of <. The 
indefinite perspectives from these points meet in 
the point /, which is the perspective of F. The 
perspective of E can be found as you did F, but 
where the construction hues intersect the picture 
plane is in a Hne parallel to the base line and 2.19 
feet above it. These points are shown at e^ e^ e^j 
and are 20.60, 36.03, 5.17 to the left of <,,. The 
perspective of G is ^ in a plane parallel to the 
base plane and 15.31 feet above it. The con- 
struction lines intersect the picture plane m 



66 ELEMENTARY COURSE IN PERSPECTIVE 

Q"y gly gl', 6.41, 13.64 feet to the left and 0.82 
feet to the right respectively of s^^. The other 
corners can be gotten in the same way and by 
connecting in proper sequence we have the per- 
spective efghi, etc., of the parallelopiped block 
E F G KL, etc. Another check on this is to find 
the vanishing points of the systems of hnes E G, 
etc. and E F, etc. These can be computed, and 
they lie directly below and above (V. P.)^ respect- 
ively. Another check is that the block rests in 
the center of the edge A D of the cube. 

PROBLEMS IN ANGULAR PERSPECTIVE 

XXII. Find the perspective of all the problems 
under angular perspective in Part I, but solve 
them all by the method of scale. 



« e.o 




iV.R)i^. 



r— 



T 




l(fp Horizon 



(V.l?)r 




4, 

J 



Chapter IX 



PERSPECTIVES OF CURVES 

XXIII. Figure 15 shows the projections of a 
cylinder with its proper location from the pic- 




Elevation 

Scale 1" = 20 ft. 
Fig. 15. 



ture and horizon planes. The cyHnder is 15 feet in 
diameter and 18 feet long. The upper base is 10 
feet below the horizon plane. The axis is parallel 

67 



68 ELEMENTARY COURSE IN PERSPECTIVE 

to the picture plane and 12.5 feet behind it. The 
station is to the right of the axis of the cyUnder 
20 feet and 18.75 feet in front of the picture plane. 
This drawing also shows the projections of two 
points, A, B and A^, B^ on each base of the cyUn- 
der, and the horizontal projections of the con- 
struction hues through these points. The points 
where these construction lines intersect the pic- 
ture plane will have to be computed. The pro- 
jections of the construction Hues through A 
intersect the picture plane in af, a^, af, through 
B in 6f, ?)f, hi, and the vanishing points in 
(V. P.)^ (V. P.)^ (V. p.)?. 

Figure 16 shows the perspective of this cylinder 
in the position shown by Fig. 15. The horizon is 
first drawn and the vertical projection s" of the 
station located. This is also the vanishing point 
V. P. The vanishing points (V. P.)^ and (V. P.), 
are also located being 18.75 feet to the left and 
right of s" or V. P. The lower base line is located 
28 feet below the horizon, and sj' is projected 
from s" upon it, and is the initial point for measur- 
ing points in the lower base line. The construc- 
tion Hues through A^ intersects the base hne in 
al, al, al, and are 25.3, 43.1, 7.5 feet to the left 
of sj". These indefinite perspectives meet in the 
point a^ which is the perspective of the point A^, 
The perspective of Bi is h^ at the intersection of 



PERSPECTIVE OF CURVES 69 

the indefinite perspectives of the construction 
hnes through B^, They intersect the lower base 
Une in h";, h^ h^, at 22.87, 28.44, 17.30 feet to the 
left of Si. The construction lines for the per- 
spectives A and B intersect the picture plane at 
the same distances to the left of s""^^, which is 
directly above s^, as those for A^ and B^ intersect 
to the left of sj', but in a plane 18 feet above the 
lower base plane, as shown at a", aj, aj, 6'', 6J, 6J. 
As a second check a is directly above a^ and h 
directly above 61, which are two elements of the 
cylinder. The perspective of the points in the 
upper base as well as in the lower base are con- 
nected by a smooth curve, and we have the per- 
spectives of the upper and the lower bases. The 
extreme elements are drawn tangent to these 
perspectives of the bases, and this completes the 
perspective aha^h^ of the cy Under AB A^B^. 

PROBLEMS IN PERSPECTIVE OF CURVES 

XXIV. Find the perspective of all the problems 
under perspective of curves in Part I, but solve 
them all by the method of scale. 



Chapter X 

PERSPECTIVE OF SHADOWS 

XXV. These are obtained by finding the per- 
spective of the horizontal projection of the rays and 
the perspective of the rays themselves and where 
they intersect is the perspective of the shadow 
of the point. This is called the method by the 
vanishing point of rays. This method has been 
shown in Fig. 7, Part I, where the vanishing 
points were obtained by drafting methods. In 
using the method by scale the vanishing points 
are located by scaling the distance that the sta- 
tion is in front of the picture plane. The van- 
ishing point of the horizontal projection of the 
rays is in the horizon, therefore their vanishing 
point (V. P.)^, Fig. 7, is found by scahng on the 
horizon to the right of the vertical projection of 
the station s"". The vanishing point of the rays 
themselves is in a Hne perpendicular to the horizon 
through (V. P.)^, Fig. 7, and below it the distance 
that the station is in front of the picture plane. 
Having obtained the perspective of a point and 
its projection on the horizontal plane that the 

70 





^ 




iV.Rh 


r/zon (V.Jt 


V 






" ~^ " " ^ ''•^ 










^ ^ 


11 \ _ r^0^^^^ "" ' -^ 


^ 


- — ' '' "^ ^.^^^^^ 1 - ' "l "^"^ -- - " "" 


y" 


^^, 






"" ^^^ 


1 ^ i 






rSir--'V--f-----; 

1 J V 1 / / 




^ 


11/ / / 




/ 
/ 


fe;>'' ^^ 




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Scale r= 6 ft. 




Fig. 16. 






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Scale 1* = 6 ft. 
Fra. 16. 



PERSPECTIVE OF SHADOWS 71 

shadow is on, we can now obtain the perspective 
of the shadow of this point as described under 
Fig. 7, Part I. 

PROBLEMS IN THE PERSPECTIVE OF 
SHADOWS 

XXVI. 1. Find the perspective of all the prob- 
lems under perspective of shadows in Part I, but 
find your vanishing points by the method of 
scale. 

2. Find the perspective of the shadow of the 
above objects on oblique planes, and on planes 
perpendicular to the horizontal plane, but obhque 
to the vertical plane. 



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WRIGHT, Prof. T. W. Elements of Mechanics, including 

Kinematics, Kinetics, and Statics. Seventh Edition, revised. 
8vo, cloth $2. 50 

■ and HAYFORD, J. F. The Adjustment of Observa- 
tions by the Method of Least Squares, with Applications to Geo- 
detic Work. Second Edition. 8vo, cloth, illustrated. . . .net, $3.00 



JAK 1319<0 



One copy del. to Cat. Div. 



